Example Problems - NN

Example-1:

Calculate the output of a 3-input neuron with X=[0,1.0,0.5], W=[0.9,0.2,0.3] and b= -0.04.Use sigmoid activation function.

Step 1: Calculate the weighted sum

The neuron first calculates:

$$z=\sum_{i=1}^{3}w_ix_i+b$$

Substituting the values:

$$z=(0.9)(0)+(0.2)(1.0)+(0.3)(0.5)-0.04$$

Calculate each term:

$$(0.9)(0)=0$$ $$(0.2)(1.0)=0.2$$ $$(0.3)(0.5)=0.15$$

Therefore,

$$z=0+0.2+0.15-0.04$$ $$\boxed{z=0.31}$$

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Step 2: Apply the sigmoid activation function

The sigmoid function is:

$$\boxed{\sigma(z)=\frac{1}{1+e^{-z}}}$$

For $z=0.31$:

$$\sigma(0.31)=\frac{1}{1+e^{-0.31}}$$

We know:

$$e^{-0.31}\approx0.7334$$

Therefore:

$$\sigma(0.31)=\frac{1}{1+0.7334}$$ $$=\frac{1}{1.7334}$$ $$\boxed{\sigma(0.31)\approx0.576}$$

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✅ Final Answer

$$\boxed{z=0.31}$$

and the neuron output is:

$$\boxed{y\approx0.576}$$

So, the output of the neuron using sigmoid activation is approximately $0.576$

Use the following Python Code 

import numpy as np


# defining the Sigmoid Function
def sigmoid (x):
    return 1/(1 + np.exp(-x))

# creating the input array
X=np.array([0,1.0,0.5])
W=np.array([0.9,0.2,0.3])
b=-0.04
o=np.dot(X,W)+b
sigmoutput=sigmoid(o)
print ('\n Output:',sigmoutput)


Output:


Output: 0.5768852611320463